Problem: Determine how many solutions exist for the system of equations. ${-5x+y = 5}$ ${-10x+2y = 10}$
Answer: Convert both equations to slope-intercept form: ${-5x+y = 5}$ $-5x{+5x} + y = 5{+5x}$ $y = 5+5x$ ${y = 5x+5}$ ${-10x+2y = 10}$ $-10x{+10x} + 2y = 10{+10x}$ $2y = 10+10x$ $y = 5+5x$ ${y = 5x+5}$ Just by looking at both equations in slope-intercept form, what can you determine? ${y = 5x+5}$ ${y = 5x+5}$ Both equations have the same slope and the same y-intercept, which means the lines would completely overlap. ${1}$ ${2}$ ${3}$ ${4}$ ${5}$ ${6}$ ${7}$ ${8}$ ${9}$ ${\llap{-}2}$ ${\llap{-}3}$ ${\llap{-}4}$ ${\llap{-}5}$ ${\llap{-}6}$ ${\llap{-}7}$ ${\llap{-}8}$ ${\llap{-}9}$ ${1}$ ${2}$ ${3}$ ${4}$ ${5}$ ${6}$ ${7}$ ${8}$ ${9}$ ${\llap{-}2}$ ${\llap{-}3}$ ${\llap{-}4}$ ${\llap{-}5}$ ${\llap{-}6}$ ${\llap{-}7}$ ${\llap{-}8}$ ${\llap{-}9}$ Since any solution of ${-5x+y = 5}$ is also a solution of ${-10x+2y = 10}$, there are infinitely many solutions.